Permutations and
Combinations
Objectives
• Calculate a permutation.
• Calculate a combination.
• Determine whether you should use a
combination or permutation to calculate
the number of outcomes.
Vocabulary
•
•
•
•
combination
permutation
with replacement
without replacement
Formulas
permutation
n!
n Pr 
(n  r )!
without replacement
and
order is important
combination
n Cr 
r n
n!
(n  r )!r !
without replacement
and
order is NOT
important
Find
Find
3
C2
3
P2
List all the combinations of
{a, b, c} when the elements are
taken two at a time.
Counting Flow Chart
A group of ten seniors, eight juniors,
five sophomores, and five freshmen
must select a committee of four.
How many committees are possible if
there must be one person from each
class on the committee?
A group of ten seniors, eight juniors,
five sophomores, and five freshmen
must select a committee of four.
How many committees are possible if
there can be any mixture of the
classes on the committee?
A group of ten seniors, eight juniors,
five sophomores, and five freshmen
must select a committee of four.
How many committees are possible if
there must be exactly two seniors on
the committee?
A 7/39 lottery requires choosing
seven of the numbers 1 through
39. How many different lottery
tickets can you choose? (Order
is not important, and numbers do
not repeat.)
Which Counting Technique?
1. What is being selected?
2. If the selected items can be repeated, use
the Fundamental Principle of Counting and
multiply the number of choices for each
category.
Which Counting Technique?
3. If there is only one category, use
 Combinations if the order of selections
does not matter – r is the number of
items to be selected from a total of n
items
 Permutation if the order of selection
does matter – r is the number of items
to be selected from a total of n items
Which Counting Technique?
4. If there is more than one category, use
the Fundamental Principle of Counting with
one box per category.
 If you are selecting one item per
category, the number in the box for
that category is the number of choices
for that category.
 If you are selecting more than one item
per category, the number in the box for
that category is found by using step 3.